D.J. Braun, On the Optimal Shape of Compressed Rotating Rod with Shear and Extensibility, International Journal of Non-linear Mechanics, vol. 43, no. 2, pp. 131–139, 2008.
This paper investigates the optimal shape of a compressed rotating rod that resists buckling when shear stress and axial extensibility are considered. Using Pontryagin’s maximum principle, the study formulates the problem as a fourth-order boundary value problem and shows that the optimal rod maintains a finite cross-sectional area at the free end—unlike predictions from classical Bernoulli–Euler theory. The work also demonstrates that traditional theory overestimates buckling loads, with the discrepancy more pronounced in optimally shaped rods than in constant cross-section rods of equal material and length.
Why it matters: Classical beam theory often neglects material extensibility and shear effects, leading to inaccurate stability predictions. This work highlights the importance of using generalized constitutive models for accurate structural optimization, with implications for engineering design where stability and material efficiency are critical.